Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps
Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps
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Collet-Eckmann 图的谱理论、zeta 函数和周期点分布
DOI：
10.1007/bf02096623
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发表时间：
1992
影响因子：
2.4
通讯作者：
T. Nowicki
中科院分区：
文献类型：
--
作者：
G. Keller;T. Nowicki
本文研究了具有负Schwarzian导数的单峰区间映射T对于某些常数K>；0和≧c>；1(c是T的临界点)满足Collet-Ekmann条件|Dtn(TC)|λKλCn。我们证明了唯一不变概率密度的指数混合性质，用中心极限和大偏差定理刻画了典型的(在勒贝格测度意义下)轨迹的长期行为
$$S_n=\Sigma_{i=0}^{n-1}f(T^i x)$$
证明了Perron Frobenius算子和相似转移算子的拟紧性，并将这些算子的孤立特征值与相应的Ruelle Zeta函数的极点联系起来.
AbstractWe study unimodal interval mapsT with negative Schwarzian derivative satisfying the Collet-Eckmann condition |DTn(Tc)|≧Kλcn for some constantsK>0 and λc>1 (c is the critical point ofT). We prove exponential mixing properties of the unique invariant probability density ofT, describe the long term behaviour of typical (in the sense of Lebesgue measure) trajectories by Central Limit and Large Deviations Theorems for partial sum processes of the form
$$S_n = \Sigma _{i = 0}^{n - 1} f(T^i x)$$
, and study the distribution of “typical” periodic orbits, also in the sense of a Central Limit Theorem and a Large Deviations Theorem.This is achieved by proving quasicompactness of the Perron Frobenius operator and of similar transfer operators for the Markov extension ofT and relating the isolated eigenvalues of these operators to the poles of the corresponding Ruelle zeta functions.